A trapezoidal type-2 fuzzy multi-criteria decision making method based on TOPSIS for supplier selection : An application in textile sector

Öz Supplier evaluation and selection includes both qualitative and quantitative criteria and it is considered as a complex Multi Criteria Decision Making (MCDM) problem. Uncertainty and impreciseness of data is an integral part of decision making process for a real life application. The fuzzy set theory allows making decisions under uncertain environment. In this paper, a trapezoidal type 2 fuzzy multicriteria decision making methods based on TOPSIS is proposed to select convenient supplier under vague information. The proposed method is applied to the supplier selection process of a textile firm in Turkey. In addition, the same problem is solved with type 1 fuzzy TOPSIS to confirm the findings of type 2 fuzzy TOPSIS. A sensitivity analysis is conducted to observe how the decision changes under different scenarios. Results show that the presented type 2 fuzzy TOPSIS method is more appropriate and effective to handle the supplier selection in uncertain environment. Tedarikçi değerlendirme ve seçimi, nitel ve nicel çok sayıda faktörün değerlendirilmesini gerektiren karmaşık birçok kriterli karar verme problemi olarak görülmektedir. Gerçek hayatta, belirsizlikler ve muğlaklık bir karar verme sürecinin ayrılmaz bir parçası olarak karşımıza çıkmaktadır. Bulanık küme teorisi, belirsizlik durumunda karar vermemize imkân sağlayan metotlardan bir tanesidir. Bu çalışmada, ikizkenar yamuk tip 2 bulanık TOPSIS yöntemi kısaca tanıtılmıştır. Tanıtılan yöntem, Türkiye’de bir tekstil firmasının tedarikçi seçimi problemine uygulanmıştır. Ayrıca, tip 2 bulanık TOPSIS yönteminin sonuçlarını desteklemek için aynı problem tip 1 bulanık TOPSIS ile de çözülmüştür. Duyarlılık analizi yapılarak önerilen çözümler farklı senaryolar altında incelenmiştir. Duyarlılık analizi sonuçlarına göre tip 2 bulanık TOPSIS daha efektif ve uygun çözümler üretmektedir.


Introduction
In recent decades, supply chain management (SCM) has taken remarkable attention in academic and business environment.The major aims of SCM are to maximize profit, improve customer relationship, reduce production costs and minimize inventory levels, and increase competitiveness.In competitive environment, supplier selection (SS) is very critical matter for firms which want to realize supply chain objectives such as competitive advantage.According to literature, the selection of the best supplier significantly decrease purchasing costs [1].It is likely that the manufacturer allocates more than sixty percent of its total sales on raw materials, parts, and components [2].Therefore, selecting the inappropriate suppliers increases operational and financial cost [3].
In the literature, SS has been addressed as a Multi Criteria Decision Making (MCDM) problem and a wide range of mathematical methods have been undertaken to provide more accurate and sufficient solutions [4].Among them, we mention genetic algorithm, artificial neural networks, data envelopment analysis, linear programming, analytic hierarchy process, and grey system theory.SS as a MCDM problem involves qualitative and quantitative criteria [4], [5].Decision-making process is to determine the best one from a given alternative sets with respect to overall judgments [6], [7].However, in many practical cases, the decision makers (DM) may be unable to assess precise numerical values to the supplier assessment in contrast to the traditional formulation of MCDM problems that human's judgments are symbolized as exact numbers.Because of the fact that some evaluation and selection criteria are qualitative and subjective in real life, it is difficult to represent preferences with numerical values for the DM [10].Fuzzy methods are effective tools dealing with uncertainty resulting from subjective human judgments [11], [41].In the classical set theory, an element cannot be in and out of a set at the same time.In contrast, fractional membership can be accepted in the fuzzy set theory [12].The current fuzzy MCDM technics are based on conventional type-1 fuzzy sets (T1FS) [56]- [59].In T1FSs, each element has a degree of membership which is described with a membership function (MF) valued in the interval [1].
Levels of uncertainty increase from numerical judgments to word and to perception, respectively [8].In real life, DMs undertake decisions in uncertain environments and conventional modeling techniques are insufficient while taking into consideration these uncertainties [8].
Recently, number of studies using MCDM with type-2 fuzzy sets (T2FSs) is rapidly growing as T1FSs are unable to cope with high uncertainty and complexity.To solve the limitations of T1FSs theory, Zadeh (1975) developed T2FS theory in 1975 as an extension of ordinary fuzzy sets [9], [11].Türkşen [34] argued that type-1 representation does not present a good approximation to verbal statements.Hence, T2FS may provide better approximation of uncertainty [8].Handling more uncertainty means making less assumption and, thus, more realistic solutions to real problem.Due to these advantages, T2FSs have potential to go beyond T1FSs [32].T2FSs are characterized by primary and secondary membership function.T2FSs can cope with uncertainty in complex systems more accurately than the T1FSs with the additional dimension of membership function.Although T2FSs are more difficult to apply than T1FSs, it is preferred by researchers to take into consideration uncertainty [12].
In particular, researchers have been applying interval T2FS theory to the field of MCDM problems.For example, Kahraman et al. [14] developed fuzzy MCDM approaches to select the most appropriate renewable energy alternatives.First they determine evaluation scores by using the analytic hierarchy process (AHP) and then they used method based on axiomatic design principles under fuzziness.The proposed methods were applied to select the most appropriate renewable energy alternative in Turkey.Chen and Lee [15] presented a new method to cope with fuzzy MCDM problems based on interval T2FSs.Chen et al. [16] proposed a novel fuzzy MCDM method based on interval T2FSs.Firstly, they proposed a novel method for ranking interval T2FSs.Then, they presented a novel technic for fuzzy MCDM based on the developed ranking method of interval T2FSs.Lou and Dong [17] developed a new methodology type-2 fuzzy neural networks.Paternain et al. [18] presented a construction method of Atanassov's intuitionistic fuzzy preference relations from the fuzzy preference relations given by experts.Wang et al. [19] addressed the MCDM problems under interval type-2 fuzzy environment, and presented an approach to cope with the situations in which the criteria values are represented by using interval T2FS.
Celik et al. [20] proposed an interval type-2 fuzzy (T2F) MCDM method based on TOPSIS and grey relationship analyzes to assess customer satisfaction at public transportation in Istanbul.Chen [21] presented a linear assignment method within the context of interval T2F numbers.The presented method is applied to the selection of a landfill site.
Chen et al. [22] developed an extended QUALIFLEX technic to solve MCDM problem in the interval T2FSs environment.The presented method was applied to a medical decision-making problem.Hu et al. [23] proposed a novel method based on possibility degree to figure out MCDM problem in the environment of interval T2FSs.The proposed method was applied to the overseas minerals investment for metals companies in China.Chen [24] developed an ELECTRE based MCDM within the environment of interval T2FSs.Kahraman et al. [10] presented an interval T2F AHP method together with a novel ranking method for T2FSs.The presented method is applied to a SS problem.Temur et al. [12] presented T2F TOPSIS approach to determine the most appropriate reverse logistics facility location.The proposed method was applied to e-waste recycling industry.Kilic and Kaya [25] developed a new T2F AHP and T2F TOPSIS methods to evaluate investment projects for development agencies in Turkey.Qin and Liu [19] presented three novel average ranking value formulas related to the interval T2F information.They define interval T2F entropy with trigonometric sine function based on the aggregation and combinatorial optimization.Celik et al. [31] presented an effective method that combines T2FSs and AHP to determine importance weights of critical success factors in humanitarian relief logistics management and evaluate them.Abdullah and Najib [33] proposed a new fuzzy analytic hierarchy process characterized by interval T2FS for linguistic variables.The presented model is applied to work safety evaluation problem.Liao and Xu [35] proposed a hesitant fuzzy VIKOR method for MCDM problem using hesitant preference information.Zouggari and Benyoucef [42] presented a twophase decision making approach for group multi-criteria supplier selection problem to integrate supplier selection process with order allocation.The first phase, suppliers are selected using fuzzy-AHP through four main criteria (Performance strategy, Quality of service, Innovation and Risk).In the second phase, via simulation based fuzzy TOPSIS; the criteria (price, quality and delivery) are evaluated for order allocation.Omurca [52] presented a hybrid method, which is consist of fuzzy c-means and rough set theory, for supplier selection, evaluation and development problem.Dogan and Aydin [53] developed the method that combines the Bayesian Networks and the Total Cost of Ownership methods for the supplier selection process.The proposed method is applied to automotive industry.Yue and Jia [54] proposed the TOPSIS method through using intuitionistic fuzzy information.Ayağ and Samanlioglu [55] developed analytic network process in the fuzzy environment.
The aim of this study is to present a trapezoidal type-1 fuzzy TOPSIS and T2F TOPSIS method for solving MCDM problem in vague information environment.The presented method is applied to a firm SS problem in which operates at textile sector in Turkey.The contribution of this paper is to present a trapezoidal type-1 fuzzy TOPSIS and type-2 fuzzy TOPSIS method for solving supplier selection problem in vague information environment in order to analyze the effect of the uncertainty level on solutions.This paper is organized as follows.Section 2 briefly reviews the concepts of type-1 fuzzy TOPSIS, interval T2FSs and T2F TOPSIS.In Section 3, a real life application for SS problem in a textile firm is conducted by using T1F TOPSIS and T2F TOPSIS.Then sensitivity analysis is made to show solutions under different conditions.Finally, conclusions are presented and point out future research in Section 4.

Type-1 fuzzy TOPSIS
The TOPSIS method was presented by Hwang and Yoon in 1981 [26].Although it has been widely utilized for decision making process, TOPSIS method is not able to deal with uncertainties.Chen [7] presented Fuzzy TOPSIS method to solve MCDM problems under uncertain environment.Here, linguistic variables are utilized by the DMs Dr (r=1,..,k) to assess the weights of the criteria and the ratings of the alternatives.Thus,  ̃ denotes the weight of the jth criteria Cj (j=1,..,m), given by the rth DM.  ̃  denotes the rating of the ith alternative Ai (i=1,…,n), with respect to criteria j, given by the rth DM.The method comprises the following steps [7]: 1.The evaluation criteria for SS process are identified by decision-makers, 2. The importance of criteria and the alternatives' ratings with respect to each criteria are estimated using Eq.(1 and 2).
Each criteria is evaluated by the DMs using linguistic variables depicted in Table 1 and alternatives are rated according to Table 2.
Table 1: Linguistic variables for the importance of the criteria [16].
7. Lastly, the closeness coefficient of each alternative is obtained as: The ranking order of alternatives can be determined based on the closeness coefficient,   .According to Chen et al. [27], using a linguistic variable to describe the current assessment status of each supplier according to its closeness coefficient may be more realistic approach.To describe the evaluation process of each supplier, the interval [0,1] is divided into five sub-intervals.Five linguistic variables for supplier assessment with respect to the sub-intervals are given in Table 3.

Interval type-2 fuzzy sets
T1FSs cannot cope with uncertainty in data since its membership grades are crisp numbers.Thus, T2FSs are introduced as an extension of T1FSs with a third dimension.The additional dimension helps in handling more uncertainties than T1FSs [28], [29].
According to John and Coupland [37] imprecision levels increase numbers, words and perceptions, respectively.Zadeh [38] presented type-2 FSs and higher-types of FSs to deal with this issue.Appropriate techniques for corresponding levels of precision of data can be illustrated as Figure 1.
Figure 1: Suitable methods according to precision levels of data [29].
In this section, some basic definitions of T2FSs are presented [36], [15].where   denotes an interval in [0,1].Furthermore, the T2F set  ̃ also can be represented as follows: ] and  shows union all acceptance u and x.
Definition 2.2: Let  ̃ be a T2FS in the universe discourse X represented by the type-2 MF   ̃.If all   ̃(, ) = 1, then  ̃ is called an interval T2FS.An interval T2FS  ̃ can be considered as a special case of a T2FS, given as following:  , ∈ {, }, and1 ≤  ≤ .

Type-2 fuzzy TOPSIS
In the most of multi-criteria decision-making problems, crisp numbers and fuzzy sets should be utilized simultaneously [25].
It is assumed that there are X alternatives, where X ={ 1 ,  2 , … .,   } and Y criteria, where Y={ 1 ,  2 , … .,   }.There are k DMs  1 ,  2 , … ., and   .The set Y of criteria can be divided into two sets Y1 and Y2, where they denote set of benefit, and cost attributes, respectively, Y1 ∩ Y2=∅ and Y1 ∪ Y2=Y.The details of the method is presented as follows [13], [12]: Step 1: Using linguistic terms and interval T2FSs (Table 4), establish the decision matrix Dk of the kth decision-maker and construct the average decision matrix D ̅ , respectively, shown as follows: Table 4: Linguistic terms and their corresponding interval T2F sets [12].
Step 2: Obtain the weighting matrix Wk of the criteria of the kth DMs and find the average weighting matrix W ̅ : where  ̃= ( ),  ̃ is an interval T2F set, 1 ≤  ≤ , 1 ≤  ≤ , 1 ≤  ≤  and k denotes the number of decisionmakers.
Step 6: Calculate the distances positive ideal solution and the negative ideal solution and find the relative degree of closeness C(xj) using the equations below: Step 7: Finally, rank the closeness scores (  ) in a descending order.Select the alternative with the highest (  ).

Type-1 fuzzy TOPSIS solutions
The computational procedure for type-1 fuzzy TOPSIS is summarized as follows: Step 1: The DMs (DM1, DM2, DM3) determine the evaluation criteria in order to evaluate suppliers.The related criteria is given in Figure 3.The DMs use the linguistic weighting variables given in Table 2.
The obtained subjective evaluations of each DM are given in Table 5.
Step 2: The DMs use the linguistic rating variables (given in Table 2) to assess the rating of alternative textile suppliers Si (i=1, 2, 3) with respect to each criterion shown in Table 6.
Table 6: Evaluation for supplier with respect to each criterion.Step 6-7-8: Determine positive ideal solution and negative ideal solution using Eqs.(12)(13)(14).Then calculate the distance of each alternative from positive ideal solution and negative ideal solution through Eqs.(15 and 16).
Finally, the closeness coefficient of each alternative is calculated using Eq. ( 17).Results can be seen in Table 12.It can be seen clearly in Table 12, according to type-1 fuzzy TOPSIS solution, the best supplier is Supplier 1.

Type-2 fuzzy TOPSIS solutions
In the first step, the importance criteria are determined by DMs using linguistic terms as Table 5 and interval T2FSs in Table 4. Decision matrix in Table 5 is composed of three alternatives Si (i=1,2,3) and seven criteria (C1, C2,…, C7) mentioned previously.In the second step, using Table 2 and Table 5, T2F weights ( ̃1 ) for the evaluation criteria are obtained given in Table 13.The next step is to determine the most appropriate supplier for the textile firm with T2FSs procedures.To do this, three DMs DMk (k=1,2,3) evaluated three alternative supplier Si (i=1,2,3) with respect to evaluation criteria (C1,…, C7), respectively.Evaluation scores of the alternatives are presented in Table 14.Based on Eqs.(20)(21)(22), T2F weighted evaluation matrix is obtained.Using Eqs.(23)(24), the ranks, Rank( ̃ ), for alternatives are obtained shown in Table 15.Then, using Table 15 and Eqs.(25 and 26), the ranks for the positive ideal and negative ideal solutions are determined given in Table 16.Table 18 shows type 1 fuzzy TOPSIS and type 2 fuzzy TOPSIS solutions in term of the closeness index.As can be seen Table 18, both methods, type 1 fuzzy TOPSIS and type 2 fuzzy TOPSIS, indicate S1 is the best supplier whereas S1 has bigger closeness index according to type 2 fuzzy TOPSIS.
Considering Table 3, the closeness index of S1 obtained with type 1 fuzzy TOPSIS indicates that S1 can be recommended with low risk, on the other hand, type 2 fuzzy TOPSIS score is classified as approved.

Sensitivity analysis
In this section, sensitivity analysis is conducted for T2F TOPSIS method to observe the effect of weight of criteria on the closeness index.To do this, firstly, the weight configurations for different cases shown in Table 19 are utilized.
According to the sensitivity analysis, as seen in Figure 4, ranking among the alternative suppliers can change due to different importance level of criteria.

Table 3 :
[27] linguistic variables for supplier assessment with respect to the sub-intervals[27].

Table 5 :
DMs' evaluations of importance of the criteria.

Table 11 :
The weighted normalize fuzzy decision matrix for textile product.

Table 12 :
The distances of suppliers from fuzzy positive and negative ideal solutions and the fuzzy closeness coefficient CCi for all suppliers.

Table 14 :
Evaluation scores of the alternatives.

Table 15 :
The ranks for the alternatives.

Table 16 :
The ranks for the positive ideal and negative ideal solutions.Using Eqs.(27 and 28), the distances from the positive ideal and negative ideal solutions are obtained in Table17.Finally, using Eqs.(29), the closeness index and the rankings results are calculated and given in Table17.According to Table17, Supplier 1 is the most appropriate supplier for textile firm.

Table 17 :
The distances of suppliers from fuzzy positive and negative ideal solutions and the fuzzy closeness coefficient CCi for all suppliers.

Table 18 :
The comparison of T1FT and T2FT solutions in term of the closeness index.